Conventionally, the error diffusion method is known as pseudo-halftoning to represent a multivalued image in binary representation (See “An Adaptive Algorithm for Spatial Gray Scale” in Society for Information Display 1975 Symposium Digest of Technical Papers, 1975, pp. 36). According to this method, assuming that a pixel of interest is P and its density is v, densities of adjacent pixels P0 to P3 of the pixel of interest P, v0 to v3, and a threshold value for binarization is T, a binarization error E in the pixel of interest P is distributed by empirically obtained weighting coefficients W0 to W3 into the adjacent pixels P0 to P3 so that a mean density is macroscopically equal to an original image density.
For example, when the value of output binary data is “o”,If v≧T holds, o=1, E=v−Vmax;If v<T holds, o=0, E=v−Vmin;  (1)
(Vmax: maximum density, Vmin: minimum density)v0=v0+E×W0;  (2)v1=v1+E×W1;  (3)v2=v2+E×W2;  (4)v3=v3+E×W3;  (5)
(Example of weighting coefficients: W0= 7/16, W1= 1/16, W2= 5/16, W3= 3/16)
Conventionally, when a multivalued image is outputted by a color ink-jet printer or the like using 4 color inks of cyan (C), magenta (M), yellow (Y) and black (K), the pseudo-halftoning is performed by using the error diffusion method or the like for each color. Regarding each color, the processing provides an excellent visual characteristic, however, regarding overlapped two or more colors, does not always provide such a excellent visual characteristic.
To solve this problem, Japanese Published Unexamined Patent Application Nos. Hei 8-279920 and Hei 11-10918 disclose halftoning to obtain an excellent visual characteristic even in overlapped two or more colors by using the error diffusion method for combination of two or more colors.
Further, Japanese Published Unexamined Patent Application No. Hei 9-139841 discloses similar improvement by performing pseudo-halftoning independently on two or more colors and then correcting output values by the sum of input values.
Especially, to reduce graininess of intermediate density area of color image, it is effective to perform image formation avoiding overlap between cyan (C) component and magenta (M) component, and for this purpose, the following method is employed.
FIG. 24 shows image formation control according to a conventional ink-jet method.
In this figure, image data is multivalue data where each density component (YMCK) of each pixel is represented as 8-bit data (0-255 gray-scale value).
Assuming that densities of C and M components of original image are C and M, densities Ct and Mt of the C and M components of pixel of interest in the multivalue color image are represented as follows.Ct=C+CerrMt=M+MerrCerr and Merr are error-diffused values of the C and M components with respect to the pixel of interest.
As shown in FIG. 24, regarding C and M image formation, 4 types of image formation controls are performed in accordance with the densities of the C and M components of the pixel of interest.    1. If the sum of (Ct+Mt) is equal to or less than a threshold value (Threshold1), i.e., the value belongs to an area (1) in FIG. 24, dot printing is not performed using C or M inks.    2. If the sum of (Ct+Mt) is greater than the threshold value (Threshold1) and the sum of (Ct+Mt) is less than another threshold value (Threshold2), and Ct>Mt holds, i.e., the value belongs to an area (2) in FIG. 24, dot printing using only the C ink is performed.    3. If the sum of (Ct+Mt) is greater than the threshold value (Threshold1) and the sum of (Ct+Mt) is less than the other threshold value (Threshold2), and Ct≦Mt holds, i.e., the value belongs to an area (3) in FIG. 24, dot printing is performed using only the M ink.    4. If the sum of (Ct+Mt) is equal to or greater than the other threshold value (Threshold2), i.e., the value belongs to an area (4) in FIG. 24, dot printing is performed using the C and M inks.
Note that Threshold1<Threshold2 holds.
However, in the above conventional art, as the image formation for the C and M components differs in accordance with the sum of the density values of the C and M components, the image formation control must be simple. If pixels where image data to be processed changes prior near a threshold value are adjacent to each other, a pixel where the C ink and the M ink overlap with each other and a pixel where these inks do not overlap with each other mixedly appear in the narrow area, and as a result, the quality of image formation is degraded.
To prevent the degradation of image quality, more complicated thresholds may be employed. However, the threshold condition processing must be more complicated, and processing time is prolonged.
Further, since the conventional threshold processing must be inevitably simple in the processing based on the sum of the density values of the C and M components, flexible processing cannot be performed without difficulty.
Further, if exclusive error diffusion is to be performed by using the sum of three components including the black (K) component, the processing becomes very complicated as represented in the following code.
 Ct=C+Cerr Mt=M+Merr Kt=K+Kerr If(Ct+Mt+Kt>Threshold1)If(Ct+Mt+Kt<Threshold2) If(Ct>Mt&&Ct>Kt)  Print C Else If(Mt>Ct&&Mt>Kt)  Print M Else  Print K Else   If(Ct+Mt+Kt<Threshold 3) If(Ct<Mt&&Ct>Kt)  Print M  Print K Else If(Mt<Ct&&Mt<Kt)  Print C  Print K Else  Print C  Print M Else  Print C  Print M  Print K
Further, in the above conventional art, the input multivalued image data is merely binarized by each color component and subjected to the error diffusion processing as the pseudo-halftoning. On the other hand, in accordance with the progress of color image printing technology by the ink-jet method, some ink-jet printers can handle multivalued image data for color image printing by drop modulation or use of same-color thick and thin inks.
Accordingly, it is desirable to apply multivalue error diffusion processing to the above ink-jet printer. However, in the multivalue error diffusion processing, as the threshold condition processing is so complicated, if the processing is applied to an actual printer, the reduction of printing speed is conceivable. For this reason, upon application of the multivalue error diffusion processing to an ink-jet printer to handle multivalued image data, a processing method capable of maintaining a high processing speed is desirable.
Further, as in the case of the above conventional art, in image formation by completely and exclusively arranging C component and M component dots, in an original image having respectively 50% C and M component, all the pixels are filled with C ink dots or M ink dots, ideally, as shown in FIG. 25A. In this state, if C-ink dot positions and M-ink dot positions are relatively shifted from each other for some reason as shown in FIG. 25B, the image has pixels where C-ink dot and M-ink dot overlap with each other (bluish pixels) and blank pixels without dot throughout most of the image.
Accordingly, in printing by an ink-jet printer using a printhead where C-ink nozzles and M-ink nozzles are arrayed in a scan direction of a carriage of the printer, a formed image periodically changes in accordance with the position of the carriage in the scan direction by variation in carriage scan speed or the like, as shown in FIGS. 25A and 25B, and it looks to a human eye that the density of corresponding area periodically changes due to the variation in probability of occurrence of blank pixels. In other words, to a human eye, the printed result appears as a low-quality image.
On the other hand, if C-ink dots and M-ink dots are independently arranged in an image formation, in an original image having respectively 50% C and M components as in the above case, blank pixels, pixels printed only with the C ink, pixels printed only with the M ink, and pixels printed with both the C and M inks are formed respectively at 25% occurrence uniformly in the formed image, ideally, as shown in FIG. 26A.
In the independent arrangement of C-ink dots and M-ink dots, a pixel to be printed only with the C ink may overlap with an adjacent pixel to be printed with the M ink, as shown in FIG. 26B, on the other hand, there is a probability that a pixel to be printed with both the C and M inks is printed with only the C ink or the M ink. Thus, the overall density change is small in comparison with the exclusive arrangement of C-ink and M-ink dots.
Accordingly, it is understood that the exclusive arrangement of C-ink and M-ink dots has a problem that the uniformity of image is degraded from intermediate to high density areas in view of a trade-off between the effect of reduction of graininess in a highlight portion and image formation accuracy. If only the highlight portion is taken into consideration, as respective dots are initially arranged sufficiently away from each other, the degradation of image quality due to shift of dot positions is very little and the advantage of the exclusive arrangement is rather greater.